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On the form-closure property of robotic grasping
Preprints of the Fourth IFAC Symposium on Robot Control September 1~21, 1994, Capri, Italy
ON THE FORM-CLOSURE PROPERTY OF ROBOTIC GRASPING A. BICCHI· ·Cent,.., "E. Piaggio", Uni"e,..itli tii Pi"., "ia Dioti.al"i, t. chithm. cnuu.cn,..it
$6115 Pi.a • Italia.
Email: I>ic.
Abstract. The form-closure and force-closure properties of robotic grasping can be loosely defined as the capability of the robot to inhibit motions of the workpiece in spite of externally applied force •. In this paper, the form-closure property of robotic grasping i. considered as a purely geometric property of a set of unilateral (contact) constraints, such as those applied on a workpiece by a mechanical fixture. The concept of partial form-closure i. introduced and discusaed in relation with other concepts appeared in literature, such as accessibility and detachability. An algorithm i. proposed to obtain a synthetic geometric description of partial form-closure constraint •. Ke), Words. Grasping, Form-Closure, Linear programming, Convex Programming.
1. INTRODUCTION
The form-closure property of constraining devices, as originally investigated by Reuleaux (1875), is related with the ability to prevent motions of the grasped object, relaying only on unilateral, frictionless contact constraints. An example of a form-closure grasp of a square object by means of four contacts is depicted in fig.1a. Contacts are represented with fixed pins at the contact points, indicating that only motions of the object that cause penetration of the pin in the object are prevented by that constraint. Reuleaux showed that at least four contact points are necessary to achieve the form-closure property in the planar case, and Somov (1900) found that at least seven are needed in the general spatial case. Lakshminarayana (1978) reports about these results and gives new proofs in the spatial case. The latter author also introduced the idea of partially restraining grasps, to take into account the more general case of devices intended to allow only some degrees-of-freedom to the object. An example of what will be later defined as a "partially form-closure" grasp is reported in fig.l-b, where only translation in the horizontal dir.ection is allowed to the object by the four contact constraints. Ohwovoriole (1980) and Salisbury (1982) introduced closure properties in the robotics literature and used screw theory for approaching the problem. The analysis of form-closure is intrinsically geometric, in so far as it does not consider the kinematics of the grasping mechanism (hand or fixture), nor the magnitude of contact forces, and it lends to rather elegant treatments mostly derived from linear and convex programming tech-
a)
b)
Fig. 1. Total and partial form-closure examples.
niques. Many contributions to form-closure study have been focused on the problem of grasp synthesis, i.e. given the object geometry, to place contacts so as to prevent object motions. The analysis problem, i.e. given an object and a set of contact locations, to decide whether the object has any degree-of-freedom left (and which), has been comparatively less thoroughly investigated so far. Qualitative (true/false) tests for formclosure have been proposed by Lakshminarayana (1978), Salisbury (1982), Nguyen (1986), Mishra et al. (1986), and Hirai and Asada (1993). On the other hand, Kirkpatrick et al. (1989) and Trinkle (1992) proposed quantitative tests, i.e., provided a quality index for the grasp under examination. After Lakshminarayana (1978) introduced it, the concept of partial form-closure has found application in the field of workpart fixturing design, where Asada and By (1985) proposed the related concepts of accessibility and detachability of workpieces. Brock (1988) considered partially 219
twise. Analogous definitions hold for the planar case, that are omitted here. In the following, d indicates the dimension of the object configuration space, i.e. d = 6 in three-dimensional space, and d = 3 in the plane.
restrained grasps from the point of view of dexterous manipulation . Trinkle (1992) uses the term "strong force-closure" to refer to what in this paper is referred to as partial form-closure, and provides a quantitative test for it. This paper, as well as the cited literature, deals only with infinitesimal (or first-order) motions, that corresponds to neglecting the effects of rolling between the surfaces in contact . This point is expanded in the final section of the paper. In the paper, form-closure related definitions and algorithms are considered , reviewing existing results in a somewhat different perspective. The most important novel contribution in this paper is perhaps the more complete geometric characterization of partially form-closure grasps , and in the related algorithm for the basis of the free motion cone. Such geometrical description might prove useful as a tool for building planning manipulation strategies that use controlled slippage for regrasping and repositioning the manipulated object.
Definition 1 A set of contact constraints is defined Form-Clo!ure if, for .all object motions u E IRei , at least one contact constraint is violated . Checking the form-closure property is usually regarded as a linear programming problem of the standard form
Maximize fTx { subject to N T GT X ~ 0
where f E IRei is an arbitrary constant vector. The existence of a feasible solution for any of these problems is a necessary and sufficient condition for negating the form- closure property of a set of contact constraints. Eq.(5) is sometimes given a physical interpretation , where f is a force/torque field (e.g., due to gravity) applied on the object, because of which the object moves if a feasible solution exists such that the mechanical work developed, fT x, is positive. This interpretation can be used for determining the object motions (see e.g. Trinkle (1992» . However, note that , strictly speaking, the concept of "force" is inessential to form-closure and can be altogether avoided in its treatment. The Reuleaux-Somov condition on the number of contacts necessary for form- closure can be easily derived from the above formulation :
2. FORM-CLQSURE Consider a rigid object 0 whose infinitesimal motion in three-dimensional space is described, in a fixed base frame, by the linear velocity v of a reference point fixed with the object and by its angular velocity '" This paper is focused on the properties of sets of constraints on the velocities of points Co, i = 1, ... , n on the surface of 0 that prevent the velocity of contact points to have components along a single direction (contact constraints). These constraints can be expressed as
Propo!ition 1 (Reuleaux-Somov). The minimum number of contacts necessary to form-restrain an object in its configuration space is d + 1.
(1)
n.
where E IR3 is the unit vector along the forbidden direction and pointing into the surface at Co. Frictionless contacts are modeled if is chosen normal to the surface at Co . The velocity (expressed in base frame) of Co can be written as
Proof: Since matrix NT G T has n rows and d columns, if n < d (or n = d and rank (N T G T ) < d) there exists x=/;O lying in the nullspace of NT GT. If otherwise n d and N T G T is invertT ible, then a solution of N G T x = TJ ~ 0 can be found as x = (NTGT)-lTJ . In both cases a feasible solution of (5) exists, hence the necessary condition for a form-closure grasp: n ~ d + 1. 0 The condition derived from the above equivalent linear programming problem differs from that proposed by Lakshminarayana (1978), who explicitly considers the case when the constraint matrix GN is full row rank. In that hypothesis, the above formulation can be reversed using the theorem of the separating hyperplane known from duality theory in linear programming (see e.g . (Gale, 1960» :
n.
=
(2) Juxtaposing n such relationships in matrix notation, one obtains
c = GTu,
(3)
NTc ~ 0
(4)
6 h were u. = (T v , '" T)T E IR , C' = ('T Cl" 3n IR , G is the so-called grasp matrix
'"
(5)
'T)T E
Cn
13 ) S( cn) ,
Propo!ition 2 (Lakshminarayana) . A set of contact constraints is form- closure if and only if its constraint matrix GN is full row rank and there exists y E !Rn , y > 0, such that GNy O.
S( ci) is the cross-product matrix for Co , and N = diag (nb ... , nn) . Note that inequality symbols when used for vectors are meant elemen-
=
220
set U C IRei if, for all object motions u E U, at least one contact constraint is violated .
Numerical routines are generally available in mathematical software packages that provide simple and efficient tests for either formulation of the problem (the search for a feasible solution actually constitutes the preliminary step of well-known linear programming algorithms such as the Simplex or Karmarkar's methods). Other equivalent formulations of the form-closure test have been derived and used in the literature to obtain efficient algorithms. Geometrically reformulated, proposition 2 requires that the convex hull of the columns of GN, CH(GN), contains a neighborhood of the origin (Mishra et al., 1986), or, equivalently, that no plane through the origin and containing any two column vectors of GN is a supporting hyperplane of CH (G N). The latter observation has been used by Chen and Burdick (1993) to derive an efficient test algorithm applicable to formclosure . In many cases, even though form-closure may not be verified, contact constraints partially restrain the motions of the object. Actually, these cases are most relevant to workpiece fixturing and dextrous manipulation . In fact, the existence of feasible motions for the workpiece in the vicinities of the designated fixturing configuration is a necessary condition for the existence of a path to bring the work piece in the fixturing position, or to remove it after machining. To characterize this problem, Asada and By (1985) introduced the property of acceuibility and detachability. In its basic form this property is equivalent to the negation of form-closure, but it has a "strong" form that requires that the object can be placed or removed from the grasp by moving in a nearby region where none of the contacts remain active. Furthermore, in many fine manipulation operations, human hands (or dextrous robotic endeffectors, see e.g. (Brock, 1988)) constrain in turn only some motions of the object while setting others free to slip, thus increasing the dexterity of manipulation. It can be expected that this type of manipulation could provide better dexterity and efficiency than techniques based on finger "gaiting" through successive statically stable configurations, such as those described by Fearing (1986) and Chen and Burdick (1993) . However, the problem of synthesizing sets of contact locations for selectively preventing and allowing slippage motions of grasped objects seems far from being satisfactorily achieved . In the following , it is our purpose to contribute to the partial form-closure analysis problem , by studying the subsets of object motions that are prevented and allowed , respectively, by a given set of contacts. Accordingly, the concept of partial form-closure is introduced:
The constrained subset Ut; is defined as the smallest set that contains every form-restrained subset of object motions. Correspondingly, let U, = {u E IReilNTGTu ~ O} (U, IRei \ Ut;) denote the free nbset of object motions.
=
Definition 3 (Asada and By, 1985) A set of contact constraints is defined acce"ible and detachable (A .D.) if U, 1 0; strongly so (S .A.D.) if the interior of Ut is not void.
In general, linear subspaces of IRei may be embedded both in Ut; and U, . The existence and maximal dimension of linear subspaces embedded in the free subset can be easily tested according to the following Proposition 3 The largest subspace U, C IRei embedded in Ut is the nullspace of N T G T .
Consider
Uo
E
Ut,
by definition of U, it holds
N T G T U O ~ o. Since U, is a subspace, it must also hold - N T G T U O ~ o. These inequalities together imply NT GT U O = O. 0
The Reuleaux-Somov condition 1 can be easily generalized to partial form-closure with respect to subspaces in Ut;: Proposition 4 The minimum number of contacts necessary to partially form-restrain an object with respect to an m-dimensional subspace is m + 1. Proof: Consider a constrained subspace U CUt;, and let the columns of the matrix U form a basis of U . Then, any u E U can be written as u Ux, x E IRm , and the above introduced equivalent linear programming problem can be modified for partial form-closure simply as
=
(6) The rest of the proof follows that of proposition 1.
o Next , the geometric structure of the free and constrained subsets of object motions in a grasp are investigated. Proposition 5 The free and constrained subsets of object motions are cones in IRei; the free subset U, is a convex polyhedral cone. Proof: From definition 2, it can be easily verified that x E Ut; implies JlX E Uc for all Jl > 0, hence Uc is a cone . Also , by its construction , the free subset is the intersection of the closed halfspaces defined by contact constraints: Ut = ni=l, .. Si,
Definition 2 A set of contact constraints is defined partially form-closure with respect to a sub-
Si 221
= {x E IReilnfGTx ~ O} , hence it is a polyhe-
,.~
dral convex cone. 0 Because they belong to a convex polyhedral cone, all free object motions can be described as a positive linear combination of a finite number of basis vectors. In- the terminology of polyhedral convex cones, this change of description is termed conTler"ion from face form to span form (Goldman and Tucker, 1956; Hirai and Asada, 1993). Related algorithms are derived from linear programming techniques. An algorithmic description of a convex basis of U" C B(U,), i.e. of a set with minimal cardinality composed of vectors that positively span the free subset, is provided in the following.
= 1, ... ,,,
"0
= (
d:
Example 1. Consider the planar grasp of the object depicted in fig.2-a. Contacts are placed at points Cl = [_IO]T; Cl = [-v'2/2 -v'2/2jT; C3 = [0 _1]T ; Col [1 O]T, and the associated directions are nl = [10]T; Cl = [10]T; C3 = [OI]T; C. = [-IO]T. Accordingly, the matrix GN is
1 ) vec-
=
GN=
Notice that the number oC vectors is larger than ~o if some oC the constraint directions are coincident. I{ m is the number oC pain oC constraints i. j such that Di aD;. a E R. then intersection operations will yield .0
~ .. -,)! + m(..+l}!(2-3}!
--
[n( n - 1 )
+ m(d -
e)
3. EXAMPLES
1
=
[!
Th~
actual number oC vect9J'8 in
. _ (,,\.. -1)(r.+1)+j,,-r.-1)(,,-r.-2 11l--2)! • ( ..-,,+ +1}!(2-'-1)! .
this
case
1
o v'2/2
0 1 0
o 1, o
-1
and is full row rank. The nullspaces of the individual contact constraints are
l)(d -
(.. -2)! 2)] ( .. -<11+ l)!( 2-1)! vectors.,,;. ,
T
Propo"ition 6 A set of contact constraints is strongly accessible and detachable only if the cardinality of Cs(U,) is greater or equal to d. Ditto if and only if the subspace spanned by vectors in C B(U,) has dimension d.
tors spanning the supporting hyperplanes l . Note that each intersection results in at least h + 1 vectors that can be linearly combined so as to have a basis of W in the first h vectors followed by vectors
--
I ....
The fact that C B(U,) has minimal cardinality entails the following
If the rank of GN is d - h, d - 1 > h > 0, then the h-dimensional nullspace W of N T G T belongs to U, . The boundary of the (not strictly) convex polyhedral cone U, is formed by Wand by (h + I)-dimensional half-hyperplanes all of which intersect in W . Form all possible intersections of the n constraint nullspaces W, d - h - 1 at a time,
•
/.
The above algorithm can be applied also to the limit case h = d-I, provided that the intersection vectors are replaced by the constraints, wf+j = nfG (see Example 3). (end of algorithm 1)
1.
~_
T
Wj rt W. Operate on the latter vectors Wj, j > h as described above, i.e., ifNTGTWj 2': 0, leave Wj unchanged; if NTGTWj :s 0, change sign to every component of Wj; if neither condition applies, discard Wj; hence take the vertices of the convex hull of the resulting set. Let WIo, k = 1, ... , h denote any set of basis vectors for W: the set of WIo'S and -WIo'S is a convex basis for W. Finally, a convex basis for U, is obtained as the union of the vertices of the convex hull of the Wj'S with the convex basis for W (see Example 2) .
1 ) vectors Wj ,
thus obtaining at least (h + 1) ( d _
I
..
Fig. 2. Example 1.
Operate on this set of vectors as follows : if NTGTWj 2': 0, leave Wj unchanged; if NTGTWj :s 0, change sign to every component of Wj; if neither condition applies, discard Wj. Finally, discard any vector in the set resulting from these operations that can be obtained as ~ convex combination of other vectors in the set (i.e., take the vertices of the convex hull of the set). The resulting set of vectors form CB(U,) (this follows from the fact that U, is a strictly convex polyhedral cone, and the w;'s lie at its edges). The algorithm above is illustrated in Example 1. j
•--:or
.)
Algorithm 1 (Basis of the Free Motion Cone.) Consider the nullspace W. of the i - th contact constraint vector nTG. Suppose at first that the constraint matrix GN is full rank, hence the number of constraints n is greater or equal to d. Form all possible intersections of the n constraint nullspaces d -1 at a time: these operations
will provide at least
•
W1 =
i•
222
Ipa n
[!
~ 1 W, = j
Ipa n
[ o~ v'2~l 1
y
Ws
[~ ~]
= .pan
; W.
= .pan
[!
"
~]. 1
The intersections of these subspaces taken (d 1 = 3 - 1 =) 2 at a time have basis vectors Wj given by
W 1 n W 2 = span W 1 n Wa = span W 2 n W3 = span W 1 n W 4 = span
W1; W2; W3; [W4
W1 = [01 O)T; W2 [00 I)T; W3 [-10 V2)T;
=
a)
=
w,);
= [01 O)T; w, = [00 I)T;
W4
W2
n W 4 = span
Wa
n W4
Fig. 3. Example 2.
= [01 O)T; = [00 I)T.
W6; W6
= span W7;
b)
W7
,,-
Condition NTGTWj ~ 0 fails only for W3, and, since also N T GTwa < 0 fails, Wa must be discarded. Therefore, C B(U,) for this example is given by the vertices of CH({W1,W2,W4,W"W6,W7}), i.e. CB(U,) {W1, W2} (see fig .2-b) . The object is accessible and detachable by means of motions in the first quadrant of the y, {) plane. However, since U, has no interior points, the grasp is not S.A.D .. If the contact point C4 is removed, the fourth column of GN is deleted, and basis vectors for the two-by-two intersections of nuIIspaces are W1, W2, and Wa . Since in this case N T G T Wa :S 0, the sign of Wa is reversed. A convex basis of U, in this case is given by CB(U,) {W1, W2, wa} (see fig .2-c), and the grasp is S.A.D .
"...... I-Jr,
-" ....
",\ )
, I
T
---. I .... -
-,,"
= .)
= =
GN
=
[!
= =
-1/2]
V3/2 ~/2
o
=
and possesses a I-dimensional nuIIspace W [OOI)T. The nuIIspaces of the individual contact constraints are
GN=
[H
0 [
~
v'~ 3]
=
-1 ]
o o
,
=
and there is a 2-dimensional nuIIspace W [01 O)T, W2 [00 I)T . Since in this example d - h - 1 0, take Wa G T0 1 [1 oOjT, W4 = G T 0 2 = [1 OO)T, T Ws G Oa [-1 0 O)T . Th is vectors are all discarded through the test N T G T Wj 2:: 0, hence we have CB(U,) {Wl,W2,-Wt.-W2} (see fig.4b) and, by proposition 6, the set of constraints is A.D . but not S.A .D .. If the contact in Ca is removed, and the third column of GN deleted, vectors Wa and W4 pass the test N T G T Wj 2:: O. The only vertex of CH({Wa,W4}) IS Wa . Therefore, CB(U,) span [W1,W2], W1
span
=
oy .
=
,
span
W3 =
=
=
=
1/2
=
such bases; let W1 [00 IjT; W2 [01 OjT; Wa [v'3 - IOjT; W4 [V3 1 O)T. Vector Wa fails condition N T G T Wj ~ 0 as weII as N T G T Wa :S 0, hence Wa is discarded. Since the vertices of CH({W2, W4}) are simply {W2, W4}, a convex basis ofU, is given by CB(U,) {W1, -W1, W2, W4} (see fig .3-b). By application of proposition 6, the grasp is S.A.D . Example 3. To iIIustrate a limit case in the application of algorithm 1, consider the example depicted in fig.4-a, where two contacts are applied at the same point Cl C2 [-1 O)T, that inhibit motions along the direction 01 02 [1 O)T, and a third contact is placed in Ca = [1 O)T with 03 [-1 In this case the grasp constraint matrix is
Example 2. For the example in fig.3-a, contacts are placed in Cl = [-10)T; C2 = [-1/2 -V3/2)T; Ca [1/2 - V3/2)T, and the associated directions are 01 [1 OjT; 02 [1/2 V3/2jT; 03 [-1/2 V3/2)T . The grasp constraint matrix is
=
c)
Fig. 4. Example 3.
=
=
1»
=
=
=
=
=
=
=
=
.
The intersections of these subspaces should be taken (d - h - 1 = 3 - 2 =) 1 at a time, therefore the w;'s coincide with the basis vectors of Wj. A basis of W already appears on the first columns of
=
223
{WI. W2. -WI.
Fearing, R. (1986) . Simplified Grasping and Manipulation with Dextrous Robot Hands. IEEE J. Robotic. and Automation, vol.!, no.,/. Gale, D. (1960) The Theory of Linear Economic Modeu, McGraw Hill. Goldman , A.J ., and Tucker, A.W. (1956).Polyhedral Convex Cones. Linear Inequalitie. and Related Sy.tem" , Annau of Mathematic. Studiu 98. Kuhn, H.W., and Tucker, A.W. (eds) . Princeton University Press. Hirai, S., and Asada, H. (1993). Kinematics and Statics of Manipulation Using the Theory of Polyhedral Convex Cones , Int. J. of Robotic. Re"earch, vo1.12, no.5. Kirkpatrick, D., Kosaraju, S.R., Mishra, B., and Yap, C-K. (1989). Quantitative Steinitz's Theorem with Applications to Multifingered Grasping, TR./60, Courant Inlt. of Mathematical Science6, New York Univ ., NY, Dept. of Computer Sciences. Lakshminarayana, K. (1978). Mechanics of Form Closure. ASME paper 78-DET-92. Mishra, B., Schwartz, J .T., and Sharir, M. (1987). On the Existence and Synthesis of Multifinger Positive Grips. TR 259, Courant In.t. of Mathematical Science", New York Univ., NY, Dept. of Computer Sciences, 1986. Also in Algorithmica 2(4) :541-558. Montana, D.J. (1992). Contact Stability for TwoFingered Grasps, IEEE Tran •. on Robotic. and Automation, vol.8, noA. Nguyen, V.D. (1986) . The synthesis of Stable Force-Closure Grasps, TR 905, MIT AI Lab. Cambridge, MA. Ohwovoriole, M.S. (1980). An Extension of Screw Theory and its Application to the Automation of Industrial Assemblies, Ph.D. The"i", Stanford Univer"ity, Dept. of Mechanical Engineering. Rimon , E. , and Burdick, J .W. (1993). Towards Planning with Force Constraints: On the Mer bility of Bodies in Contact, Proc. IEEE Int. Con/. on Robotic" and Automation. Reuleaux, F. (1875) . Kinematic. of Machinery. New York: Dover, 1963. First published in German. Salisbury, J .K. (1982) . Kinematic and Force Analysis of Articulated Hands, Rep. STAN-CS-8291, Stanford Univ .. Stanford , CA . Somov , P. (1900) . Uber Gebiete von Schraubengeschwindigkeiten eines starren Korpers bei verschiedener Zahl von Stutzflachen. Zeit"chri/t fUr Mathematik und Phytlik, Vol.45 . Trinkle, J.C. (1992). On the Stability and Instantaneous Velocity of Grasped Frictionless Objects, IEEE Tran". on Robotic. and A utomation, vol.8, no.5.
-w2. wa} (see fig A-c) , and the
grasp is S.A.D ..
4. CONCLUSIONS ~his paper reports on a systematic investigation on the form-closure property of grasping. In particular, the concept of partial form-closure has been studied in more depth than it had been previously, and an algorithm for describing the geometry of partial form-closure grasps has been presented. Results reported in this paper leave a n~mber of open problems in the analysis and synthesis of cler sure grasps. In particular, as already noted, these results are only valid to the first-order, and the effects of rolling of finite-curvature fingers on the object are not taken into account. The role played by higher-order properties of the surfaces in contact has been studied by Montana (1992) and applied to closure analysis by Rimon and Burdick (1993). Results presented so far are rather complex from a computational point of view, and further analysis is needed to provide algorithms suitable for implementation in a manipulation planning system. Due to the comparative simplicity of first-order analysis, and to the fact that it prer vides conservative answers to closure tests (firstorder closure implies closure), the role of firstorder tests is probably to hold a central place in grasp planning. Another direction left unexplored by this paper is how information about the mer bility of the object under partial closure grasps can be exploited for planning dexterous manipulations.
ACKNOWLEDG EMENTS The research reported in this paper has been partially supported by the C.N .R. - Progetto Finalizzato Robotica under contract 93.01079 .PF67 and 93.01047.PF67. 5. REFERENCES Asada, H., and By, A.B. (1985). Kinematic Analysis of Workpart Fixturing for Flexible Assembly with Automatically Reconfigurable Fixtures , IEEE Tran". on Robotic" and A utomation, vol.1, no.2. Brock, D.L. (1988). Enhancing the dexterity of a Robot hand Using Controlled Slip, Proc. IEEE 1nl. Conf. on Robotic" and Automation. Chen, I-M ., and Burdick, J .W. (1993). A Qualitative Test for N-Finger Force-Closure Grasps on Planar Objects with Applications to Manipulation and Finger Gaits, Proc. IEEE Int. Conf, on Robotic" and Automation.
224
ON THE FORM-CLOSURE PROPERTY OF ROBOTIC GRASPING A. BICCHI· ·Cent,.., "E. Piaggio", Uni"e,..itli tii Pi"., "ia Dioti.al"i, t. chithm. cnuu.cn,..it
$6115 Pi.a • Italia.
Email: I>ic.
Abstract. The form-closure and force-closure properties of robotic grasping can be loosely defined as the capability of the robot to inhibit motions of the workpiece in spite of externally applied force •. In this paper, the form-closure property of robotic grasping i. considered as a purely geometric property of a set of unilateral (contact) constraints, such as those applied on a workpiece by a mechanical fixture. The concept of partial form-closure i. introduced and discusaed in relation with other concepts appeared in literature, such as accessibility and detachability. An algorithm i. proposed to obtain a synthetic geometric description of partial form-closure constraint •. Ke), Words. Grasping, Form-Closure, Linear programming, Convex Programming.
1. INTRODUCTION
The form-closure property of constraining devices, as originally investigated by Reuleaux (1875), is related with the ability to prevent motions of the grasped object, relaying only on unilateral, frictionless contact constraints. An example of a form-closure grasp of a square object by means of four contacts is depicted in fig.1a. Contacts are represented with fixed pins at the contact points, indicating that only motions of the object that cause penetration of the pin in the object are prevented by that constraint. Reuleaux showed that at least four contact points are necessary to achieve the form-closure property in the planar case, and Somov (1900) found that at least seven are needed in the general spatial case. Lakshminarayana (1978) reports about these results and gives new proofs in the spatial case. The latter author also introduced the idea of partially restraining grasps, to take into account the more general case of devices intended to allow only some degrees-of-freedom to the object. An example of what will be later defined as a "partially form-closure" grasp is reported in fig.l-b, where only translation in the horizontal dir.ection is allowed to the object by the four contact constraints. Ohwovoriole (1980) and Salisbury (1982) introduced closure properties in the robotics literature and used screw theory for approaching the problem. The analysis of form-closure is intrinsically geometric, in so far as it does not consider the kinematics of the grasping mechanism (hand or fixture), nor the magnitude of contact forces, and it lends to rather elegant treatments mostly derived from linear and convex programming tech-
a)
b)
Fig. 1. Total and partial form-closure examples.
niques. Many contributions to form-closure study have been focused on the problem of grasp synthesis, i.e. given the object geometry, to place contacts so as to prevent object motions. The analysis problem, i.e. given an object and a set of contact locations, to decide whether the object has any degree-of-freedom left (and which), has been comparatively less thoroughly investigated so far. Qualitative (true/false) tests for formclosure have been proposed by Lakshminarayana (1978), Salisbury (1982), Nguyen (1986), Mishra et al. (1986), and Hirai and Asada (1993). On the other hand, Kirkpatrick et al. (1989) and Trinkle (1992) proposed quantitative tests, i.e., provided a quality index for the grasp under examination. After Lakshminarayana (1978) introduced it, the concept of partial form-closure has found application in the field of workpart fixturing design, where Asada and By (1985) proposed the related concepts of accessibility and detachability of workpieces. Brock (1988) considered partially 219
twise. Analogous definitions hold for the planar case, that are omitted here. In the following, d indicates the dimension of the object configuration space, i.e. d = 6 in three-dimensional space, and d = 3 in the plane.
restrained grasps from the point of view of dexterous manipulation . Trinkle (1992) uses the term "strong force-closure" to refer to what in this paper is referred to as partial form-closure, and provides a quantitative test for it. This paper, as well as the cited literature, deals only with infinitesimal (or first-order) motions, that corresponds to neglecting the effects of rolling between the surfaces in contact . This point is expanded in the final section of the paper. In the paper, form-closure related definitions and algorithms are considered , reviewing existing results in a somewhat different perspective. The most important novel contribution in this paper is perhaps the more complete geometric characterization of partially form-closure grasps , and in the related algorithm for the basis of the free motion cone. Such geometrical description might prove useful as a tool for building planning manipulation strategies that use controlled slippage for regrasping and repositioning the manipulated object.
Definition 1 A set of contact constraints is defined Form-Clo!ure if, for .all object motions u E IRei , at least one contact constraint is violated . Checking the form-closure property is usually regarded as a linear programming problem of the standard form
Maximize fTx { subject to N T GT X ~ 0
where f E IRei is an arbitrary constant vector. The existence of a feasible solution for any of these problems is a necessary and sufficient condition for negating the form- closure property of a set of contact constraints. Eq.(5) is sometimes given a physical interpretation , where f is a force/torque field (e.g., due to gravity) applied on the object, because of which the object moves if a feasible solution exists such that the mechanical work developed, fT x, is positive. This interpretation can be used for determining the object motions (see e.g. Trinkle (1992» . However, note that , strictly speaking, the concept of "force" is inessential to form-closure and can be altogether avoided in its treatment. The Reuleaux-Somov condition on the number of contacts necessary for form- closure can be easily derived from the above formulation :
2. FORM-CLQSURE Consider a rigid object 0 whose infinitesimal motion in three-dimensional space is described, in a fixed base frame, by the linear velocity v of a reference point fixed with the object and by its angular velocity '" This paper is focused on the properties of sets of constraints on the velocities of points Co, i = 1, ... , n on the surface of 0 that prevent the velocity of contact points to have components along a single direction (contact constraints). These constraints can be expressed as
Propo!ition 1 (Reuleaux-Somov). The minimum number of contacts necessary to form-restrain an object in its configuration space is d + 1.
(1)
n.
where E IR3 is the unit vector along the forbidden direction and pointing into the surface at Co. Frictionless contacts are modeled if is chosen normal to the surface at Co . The velocity (expressed in base frame) of Co can be written as
Proof: Since matrix NT G T has n rows and d columns, if n < d (or n = d and rank (N T G T ) < d) there exists x=/;O lying in the nullspace of NT GT. If otherwise n d and N T G T is invertT ible, then a solution of N G T x = TJ ~ 0 can be found as x = (NTGT)-lTJ . In both cases a feasible solution of (5) exists, hence the necessary condition for a form-closure grasp: n ~ d + 1. 0 The condition derived from the above equivalent linear programming problem differs from that proposed by Lakshminarayana (1978), who explicitly considers the case when the constraint matrix GN is full row rank. In that hypothesis, the above formulation can be reversed using the theorem of the separating hyperplane known from duality theory in linear programming (see e.g . (Gale, 1960» :
n.
=
(2) Juxtaposing n such relationships in matrix notation, one obtains
c = GTu,
(3)
NTc ~ 0
(4)
6 h were u. = (T v , '" T)T E IR , C' = ('T Cl" 3n IR , G is the so-called grasp matrix
'"
(5)
'T)T E
Cn
13 ) S( cn) ,
Propo!ition 2 (Lakshminarayana) . A set of contact constraints is form- closure if and only if its constraint matrix GN is full row rank and there exists y E !Rn , y > 0, such that GNy O.
S( ci) is the cross-product matrix for Co , and N = diag (nb ... , nn) . Note that inequality symbols when used for vectors are meant elemen-
=
220
set U C IRei if, for all object motions u E U, at least one contact constraint is violated .
Numerical routines are generally available in mathematical software packages that provide simple and efficient tests for either formulation of the problem (the search for a feasible solution actually constitutes the preliminary step of well-known linear programming algorithms such as the Simplex or Karmarkar's methods). Other equivalent formulations of the form-closure test have been derived and used in the literature to obtain efficient algorithms. Geometrically reformulated, proposition 2 requires that the convex hull of the columns of GN, CH(GN), contains a neighborhood of the origin (Mishra et al., 1986), or, equivalently, that no plane through the origin and containing any two column vectors of GN is a supporting hyperplane of CH (G N). The latter observation has been used by Chen and Burdick (1993) to derive an efficient test algorithm applicable to formclosure . In many cases, even though form-closure may not be verified, contact constraints partially restrain the motions of the object. Actually, these cases are most relevant to workpiece fixturing and dextrous manipulation . In fact, the existence of feasible motions for the workpiece in the vicinities of the designated fixturing configuration is a necessary condition for the existence of a path to bring the work piece in the fixturing position, or to remove it after machining. To characterize this problem, Asada and By (1985) introduced the property of acceuibility and detachability. In its basic form this property is equivalent to the negation of form-closure, but it has a "strong" form that requires that the object can be placed or removed from the grasp by moving in a nearby region where none of the contacts remain active. Furthermore, in many fine manipulation operations, human hands (or dextrous robotic endeffectors, see e.g. (Brock, 1988)) constrain in turn only some motions of the object while setting others free to slip, thus increasing the dexterity of manipulation. It can be expected that this type of manipulation could provide better dexterity and efficiency than techniques based on finger "gaiting" through successive statically stable configurations, such as those described by Fearing (1986) and Chen and Burdick (1993) . However, the problem of synthesizing sets of contact locations for selectively preventing and allowing slippage motions of grasped objects seems far from being satisfactorily achieved . In the following , it is our purpose to contribute to the partial form-closure analysis problem , by studying the subsets of object motions that are prevented and allowed , respectively, by a given set of contacts. Accordingly, the concept of partial form-closure is introduced:
The constrained subset Ut; is defined as the smallest set that contains every form-restrained subset of object motions. Correspondingly, let U, = {u E IReilNTGTu ~ O} (U, IRei \ Ut;) denote the free nbset of object motions.
=
Definition 3 (Asada and By, 1985) A set of contact constraints is defined acce"ible and detachable (A .D.) if U, 1 0; strongly so (S .A.D.) if the interior of Ut is not void.
In general, linear subspaces of IRei may be embedded both in Ut; and U, . The existence and maximal dimension of linear subspaces embedded in the free subset can be easily tested according to the following Proposition 3 The largest subspace U, C IRei embedded in Ut is the nullspace of N T G T .
Consider
Uo
E
Ut,
by definition of U, it holds
N T G T U O ~ o. Since U, is a subspace, it must also hold - N T G T U O ~ o. These inequalities together imply NT GT U O = O. 0
The Reuleaux-Somov condition 1 can be easily generalized to partial form-closure with respect to subspaces in Ut;: Proposition 4 The minimum number of contacts necessary to partially form-restrain an object with respect to an m-dimensional subspace is m + 1. Proof: Consider a constrained subspace U CUt;, and let the columns of the matrix U form a basis of U . Then, any u E U can be written as u Ux, x E IRm , and the above introduced equivalent linear programming problem can be modified for partial form-closure simply as
=
(6) The rest of the proof follows that of proposition 1.
o Next , the geometric structure of the free and constrained subsets of object motions in a grasp are investigated. Proposition 5 The free and constrained subsets of object motions are cones in IRei; the free subset U, is a convex polyhedral cone. Proof: From definition 2, it can be easily verified that x E Ut; implies JlX E Uc for all Jl > 0, hence Uc is a cone . Also , by its construction , the free subset is the intersection of the closed halfspaces defined by contact constraints: Ut = ni=l, .. Si,
Definition 2 A set of contact constraints is defined partially form-closure with respect to a sub-
Si 221
= {x E IReilnfGTx ~ O} , hence it is a polyhe-
,.~
dral convex cone. 0 Because they belong to a convex polyhedral cone, all free object motions can be described as a positive linear combination of a finite number of basis vectors. In- the terminology of polyhedral convex cones, this change of description is termed conTler"ion from face form to span form (Goldman and Tucker, 1956; Hirai and Asada, 1993). Related algorithms are derived from linear programming techniques. An algorithmic description of a convex basis of U" C B(U,), i.e. of a set with minimal cardinality composed of vectors that positively span the free subset, is provided in the following.
= 1, ... ,,,
"0
= (
d:
Example 1. Consider the planar grasp of the object depicted in fig.2-a. Contacts are placed at points Cl = [_IO]T; Cl = [-v'2/2 -v'2/2jT; C3 = [0 _1]T ; Col [1 O]T, and the associated directions are nl = [10]T; Cl = [10]T; C3 = [OI]T; C. = [-IO]T. Accordingly, the matrix GN is
1 ) vec-
=
GN=
Notice that the number oC vectors is larger than ~o if some oC the constraint directions are coincident. I{ m is the number oC pain oC constraints i. j such that Di aD;. a E R. then intersection operations will yield .0
~ .. -,)! + m(..+l}!(2-3}!
--
[n( n - 1 )
+ m(d -
e)
3. EXAMPLES
1
=
[!
Th~
actual number oC vect9J'8 in
. _ (,,\.. -1)(r.+1)+j,,-r.-1)(,,-r.-2 11l--2)! • ( ..-,,+ +1}!(2-'-1)! .
this
case
1
o v'2/2
0 1 0
o 1, o
-1
and is full row rank. The nullspaces of the individual contact constraints are
l)(d -
(.. -2)! 2)] ( .. -<11+ l)!( 2-1)! vectors.,,;. ,
T
Propo"ition 6 A set of contact constraints is strongly accessible and detachable only if the cardinality of Cs(U,) is greater or equal to d. Ditto if and only if the subspace spanned by vectors in C B(U,) has dimension d.
tors spanning the supporting hyperplanes l . Note that each intersection results in at least h + 1 vectors that can be linearly combined so as to have a basis of W in the first h vectors followed by vectors
--
I ....
The fact that C B(U,) has minimal cardinality entails the following
If the rank of GN is d - h, d - 1 > h > 0, then the h-dimensional nullspace W of N T G T belongs to U, . The boundary of the (not strictly) convex polyhedral cone U, is formed by Wand by (h + I)-dimensional half-hyperplanes all of which intersect in W . Form all possible intersections of the n constraint nullspaces W, d - h - 1 at a time,
•
/.
The above algorithm can be applied also to the limit case h = d-I, provided that the intersection vectors are replaced by the constraints, wf+j = nfG (see Example 3). (end of algorithm 1)
1.
~_
T
Wj rt W. Operate on the latter vectors Wj, j > h as described above, i.e., ifNTGTWj 2': 0, leave Wj unchanged; if NTGTWj :s 0, change sign to every component of Wj; if neither condition applies, discard Wj; hence take the vertices of the convex hull of the resulting set. Let WIo, k = 1, ... , h denote any set of basis vectors for W: the set of WIo'S and -WIo'S is a convex basis for W. Finally, a convex basis for U, is obtained as the union of the vertices of the convex hull of the Wj'S with the convex basis for W (see Example 2) .
1 ) vectors Wj ,
thus obtaining at least (h + 1) ( d _
I
..
Fig. 2. Example 1.
Operate on this set of vectors as follows : if NTGTWj 2': 0, leave Wj unchanged; if NTGTWj :s 0, change sign to every component of Wj; if neither condition applies, discard Wj. Finally, discard any vector in the set resulting from these operations that can be obtained as ~ convex combination of other vectors in the set (i.e., take the vertices of the convex hull of the set). The resulting set of vectors form CB(U,) (this follows from the fact that U, is a strictly convex polyhedral cone, and the w;'s lie at its edges). The algorithm above is illustrated in Example 1. j
•--:or
.)
Algorithm 1 (Basis of the Free Motion Cone.) Consider the nullspace W. of the i - th contact constraint vector nTG. Suppose at first that the constraint matrix GN is full rank, hence the number of constraints n is greater or equal to d. Form all possible intersections of the n constraint nullspaces d -1 at a time: these operations
will provide at least
•
W1 =
i•
222
Ipa n
[!
~ 1 W, = j
Ipa n
[ o~ v'2~l 1
y
Ws
[~ ~]
= .pan
; W.
= .pan
[!
"
~]. 1
The intersections of these subspaces taken (d 1 = 3 - 1 =) 2 at a time have basis vectors Wj given by
W 1 n W 2 = span W 1 n Wa = span W 2 n W3 = span W 1 n W 4 = span
W1; W2; W3; [W4
W1 = [01 O)T; W2 [00 I)T; W3 [-10 V2)T;
=
a)
=
w,);
= [01 O)T; w, = [00 I)T;
W4
W2
n W 4 = span
Wa
n W4
Fig. 3. Example 2.
= [01 O)T; = [00 I)T.
W6; W6
= span W7;
b)
W7
,,-
Condition NTGTWj ~ 0 fails only for W3, and, since also N T GTwa < 0 fails, Wa must be discarded. Therefore, C B(U,) for this example is given by the vertices of CH({W1,W2,W4,W"W6,W7}), i.e. CB(U,) {W1, W2} (see fig .2-b) . The object is accessible and detachable by means of motions in the first quadrant of the y, {) plane. However, since U, has no interior points, the grasp is not S.A.D .. If the contact point C4 is removed, the fourth column of GN is deleted, and basis vectors for the two-by-two intersections of nuIIspaces are W1, W2, and Wa . Since in this case N T G T Wa :S 0, the sign of Wa is reversed. A convex basis of U, in this case is given by CB(U,) {W1, W2, wa} (see fig .2-c), and the grasp is S.A.D .
"...... I-Jr,
-" ....
",\ )
, I
T
---. I .... -
-,,"
= .)
= =
GN
=
[!
= =
-1/2]
V3/2 ~/2
o
=
and possesses a I-dimensional nuIIspace W [OOI)T. The nuIIspaces of the individual contact constraints are
GN=
[H
0 [
~
v'~ 3]
=
-1 ]
o o
,
=
and there is a 2-dimensional nuIIspace W [01 O)T, W2 [00 I)T . Since in this example d - h - 1 0, take Wa G T0 1 [1 oOjT, W4 = G T 0 2 = [1 OO)T, T Ws G Oa [-1 0 O)T . Th is vectors are all discarded through the test N T G T Wj 2:: 0, hence we have CB(U,) {Wl,W2,-Wt.-W2} (see fig.4b) and, by proposition 6, the set of constraints is A.D . but not S.A .D .. If the contact in Ca is removed, and the third column of GN deleted, vectors Wa and W4 pass the test N T G T Wj 2:: O. The only vertex of CH({Wa,W4}) IS Wa . Therefore, CB(U,) span [W1,W2], W1
span
=
oy .
=
,
span
W3 =
=
=
=
1/2
=
such bases; let W1 [00 IjT; W2 [01 OjT; Wa [v'3 - IOjT; W4 [V3 1 O)T. Vector Wa fails condition N T G T Wj ~ 0 as weII as N T G T Wa :S 0, hence Wa is discarded. Since the vertices of CH({W2, W4}) are simply {W2, W4}, a convex basis ofU, is given by CB(U,) {W1, -W1, W2, W4} (see fig .3-b). By application of proposition 6, the grasp is S.A.D . Example 3. To iIIustrate a limit case in the application of algorithm 1, consider the example depicted in fig.4-a, where two contacts are applied at the same point Cl C2 [-1 O)T, that inhibit motions along the direction 01 02 [1 O)T, and a third contact is placed in Ca = [1 O)T with 03 [-1 In this case the grasp constraint matrix is
Example 2. For the example in fig.3-a, contacts are placed in Cl = [-10)T; C2 = [-1/2 -V3/2)T; Ca [1/2 - V3/2)T, and the associated directions are 01 [1 OjT; 02 [1/2 V3/2jT; 03 [-1/2 V3/2)T . The grasp constraint matrix is
=
c)
Fig. 4. Example 3.
=
=
1»
=
=
=
=
=
=
=
=
.
The intersections of these subspaces should be taken (d - h - 1 = 3 - 2 =) 1 at a time, therefore the w;'s coincide with the basis vectors of Wj. A basis of W already appears on the first columns of
=
223
{WI. W2. -WI.
Fearing, R. (1986) . Simplified Grasping and Manipulation with Dextrous Robot Hands. IEEE J. Robotic. and Automation, vol.!, no.,/. Gale, D. (1960) The Theory of Linear Economic Modeu, McGraw Hill. Goldman , A.J ., and Tucker, A.W. (1956).Polyhedral Convex Cones. Linear Inequalitie. and Related Sy.tem" , Annau of Mathematic. Studiu 98. Kuhn, H.W., and Tucker, A.W. (eds) . Princeton University Press. Hirai, S., and Asada, H. (1993). Kinematics and Statics of Manipulation Using the Theory of Polyhedral Convex Cones , Int. J. of Robotic. Re"earch, vo1.12, no.5. Kirkpatrick, D., Kosaraju, S.R., Mishra, B., and Yap, C-K. (1989). Quantitative Steinitz's Theorem with Applications to Multifingered Grasping, TR./60, Courant Inlt. of Mathematical Science6, New York Univ ., NY, Dept. of Computer Sciences. Lakshminarayana, K. (1978). Mechanics of Form Closure. ASME paper 78-DET-92. Mishra, B., Schwartz, J .T., and Sharir, M. (1987). On the Existence and Synthesis of Multifinger Positive Grips. TR 259, Courant In.t. of Mathematical Science", New York Univ., NY, Dept. of Computer Sciences, 1986. Also in Algorithmica 2(4) :541-558. Montana, D.J. (1992). Contact Stability for TwoFingered Grasps, IEEE Tran •. on Robotic. and Automation, vol.8, noA. Nguyen, V.D. (1986) . The synthesis of Stable Force-Closure Grasps, TR 905, MIT AI Lab. Cambridge, MA. Ohwovoriole, M.S. (1980). An Extension of Screw Theory and its Application to the Automation of Industrial Assemblies, Ph.D. The"i", Stanford Univer"ity, Dept. of Mechanical Engineering. Rimon , E. , and Burdick, J .W. (1993). Towards Planning with Force Constraints: On the Mer bility of Bodies in Contact, Proc. IEEE Int. Con/. on Robotic" and Automation. Reuleaux, F. (1875) . Kinematic. of Machinery. New York: Dover, 1963. First published in German. Salisbury, J .K. (1982) . Kinematic and Force Analysis of Articulated Hands, Rep. STAN-CS-8291, Stanford Univ .. Stanford , CA . Somov , P. (1900) . Uber Gebiete von Schraubengeschwindigkeiten eines starren Korpers bei verschiedener Zahl von Stutzflachen. Zeit"chri/t fUr Mathematik und Phytlik, Vol.45 . Trinkle, J.C. (1992). On the Stability and Instantaneous Velocity of Grasped Frictionless Objects, IEEE Tran". on Robotic. and A utomation, vol.8, no.5.
-w2. wa} (see fig A-c) , and the
grasp is S.A.D ..
4. CONCLUSIONS ~his paper reports on a systematic investigation on the form-closure property of grasping. In particular, the concept of partial form-closure has been studied in more depth than it had been previously, and an algorithm for describing the geometry of partial form-closure grasps has been presented. Results reported in this paper leave a n~mber of open problems in the analysis and synthesis of cler sure grasps. In particular, as already noted, these results are only valid to the first-order, and the effects of rolling of finite-curvature fingers on the object are not taken into account. The role played by higher-order properties of the surfaces in contact has been studied by Montana (1992) and applied to closure analysis by Rimon and Burdick (1993). Results presented so far are rather complex from a computational point of view, and further analysis is needed to provide algorithms suitable for implementation in a manipulation planning system. Due to the comparative simplicity of first-order analysis, and to the fact that it prer vides conservative answers to closure tests (firstorder closure implies closure), the role of firstorder tests is probably to hold a central place in grasp planning. Another direction left unexplored by this paper is how information about the mer bility of the object under partial closure grasps can be exploited for planning dexterous manipulations.
ACKNOWLEDG EMENTS The research reported in this paper has been partially supported by the C.N .R. - Progetto Finalizzato Robotica under contract 93.01079 .PF67 and 93.01047.PF67. 5. REFERENCES Asada, H., and By, A.B. (1985). Kinematic Analysis of Workpart Fixturing for Flexible Assembly with Automatically Reconfigurable Fixtures , IEEE Tran". on Robotic" and A utomation, vol.1, no.2. Brock, D.L. (1988). Enhancing the dexterity of a Robot hand Using Controlled Slip, Proc. IEEE 1nl. Conf. on Robotic" and Automation. Chen, I-M ., and Burdick, J .W. (1993). A Qualitative Test for N-Finger Force-Closure Grasps on Planar Objects with Applications to Manipulation and Finger Gaits, Proc. IEEE Int. Conf, on Robotic" and Automation.
224